How does scarcity affect product pricing?

What happens to the price of a product when it is scarce but in demand?

This is a pretty fundamental economic question, but the answer actually has some interesting nuances, so it’s not quite as simple as it seems. Generally, though, the “supply and demand” theory states that both as the supply for a product increases, the price decreases, and as the demand increases, the price increases.
An example of this is with art. When an artist’s work is indistinguishable from other artists, it’s in high supply. Think “Starving Artists” and those cheesy weekend sales set up at hotels in big cities. By contrast, when an artist becomes well known, suddenly their work isn’t interchangeable with others and the speed at which they can produce works becomes critically important. If they’re dead, even the better (at least, from an economic perspective!) because the supply is now fixed and each item sold effectively removes it from the market. Supply goes down and price goes up. That’s why an original Dali or Chagall will cost more than the nicest house in your town.
Demand is the other side of this equation worth considering, and generally the more demand there is for something, the greater its price. Gas is a splendid example of this: if everyone just bought 10% less gasoline, the price of gasoline would drop. Art is again an example of this too, and as an artist becomes more famous their works can dramatically increase in value. Indeed, that’s one reason why some investors believe that investing in “unknown” artists is a great way to make money in the long term.
The wrinkle with all of this is the issue of commoditization. When a product becomes a commodity, then there’s essentially infinite supply and the cost of production really becomes irrelevant to the price it can command in the marketplace. Personal computers are the perfect example of this (as my co-author Linda Sanford and I write about in detail in our book Let Go To Grow) and it’s proven an extraordinary challenge for companies like Dell (Nasdaq: DELL) and Gateway (NYSE: GTW) as they’ve seen their entire business model crumble. Computers can be so cheaply produced and built that they end up differentiating on weird things like the color of the case since every machine has the same hard drive, same memory, same CPU, and same plugs and capabilities.
Anyway, long answer to a short question. In most “well behaved” markets, the price of a product goes up dramatically when it’s been pushed by both its scarcity and demand.

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I do have a lot to say, and questions of my own for that matter, but first I'd like to say thank you, Dave, for all your helpful information by
buying you a cup of coffee!

The core goal of advertising is to increase demand for a product. If the advertising is successful, then the demand increases and the price can commensurately increase, with (ideally) the profit from the increased demand and price more than paying for the cost of the advertising.